A probability student wonders if independence can be presumed when P(AnB) = P(A)P
(B), or visually represented in a Venn diagram. Doctor Peterson clarifies that
independent events need not result from separate actions, nor must events combined
by "or" result from the same action.

Anne, Bob, and Carmel are going to take turns rolling a fair die, in
the order Anne, Bob, Carmel, Anne, Bob Carmel, etc. The first person
to roll a 6 will win $100.
a) Find the probability that Anne will win $100 if the first four
numbers rolled are 3, 2, 5, 5.
b) Find the probability that Anne will win $100 if a draw is to be
declared with nobody winning $100 in the event of a 6 not being
rolled within the first 8 rolls in total.
c) Suppose that a draw is to be declared if ever two 1's are rolled
in a row (by two different players). Also suppose that the game is
in fact over and did not end in a draw. Find the probability that
Anne is the player who won $100.

Each of 16 prisoners receives a hat that is either red or blue (the
colour is selected randomly; each has a 1/2 probability). All the
prisoners must simultaneously either try to guess the colour of their
hats, or pass.

Suppose we roll one six-sided die. What are the possible outcomes?
What is the probabiliy of rolling a 4? If we have two dice how many
outcomes are there? With two dice what is the probability of rolling a
5?

A, B, and C keep shooting in the sequence: A,B,C,A,B,C... until a target
is hit. A hits the target 5/6 of the time, B 3/4 of the time, and C 2/3
of the time. What's the probability that C will hit the target first?

There are six choices for a color I'm thinking of. Someone guessed my
color correctly, and then guessed again. She guessed correctly the second
time and did this again, and again, and again, and again. What is the
probability of this happening - that someone could guess correctly in a
row?

While all 3.4 million secretaries were at the annual secretaries' day
picnic, a vicious storm began. Lightning headed directly for the
secretaries. Given that males are eight times as likely to be hit by
lightning as females, what is the probability that one of the male
secretaries will be struck?

Find the probability of getting at least 20 duplicate addresses when
drawing a sample of 30,000 at random from UK households (estimate
21,000,000) where there is replacement every time a selection is made.